Stability Criteria for Distributed Nonlinear Sampled-Data Systems Defined by Green’s Functions

Sampled-data, nonlinear, distributed systems, which exhibit a structure similar to that of the standard closed loop with lumped parameter, are investigated from the viewpoint of their input-output stability. These systems are governed by operational equations involving discrete Laplace-Green kernels. Their feedback gains are bounded by upper and lower values which depend explicitly on the time and the distributed parameter. The main result is: an input-output stability theorem is given which applies both in L_∞ (O, ∞) and L₂ (O, ∞). This criterion, which may be considered as being an extension of the ≪circle criterion≫, involves the mean square value on the bounds of the feedback gain. Stability conditions for continuous systems are derived from this result. In the special case of systems with distributed periodical time-varying feedback gains, a stability criterion is given which applies in Marcinkiewicz space M₂ (O, ∞). This result which involves the mean square value of the feedback gain is generally less restrictive than the L₂ (O, ∞) stability criterion mentioned above.